 Solution

Write $x=x_0+y$, so the fluctuation has homogeneous Dirichlet boundary conditions $y(0)=y(T_E)=0$. Because ${\dot{x}}_0$ is constant, the cross term in the Euclidean action vanishes exactly:ConsequentlyTranslation of each intermediate integration variable has Jacobian determinant one in the time-sliced configuration-space path integral. This is an exact classical-path factorization of a quadratic path integral, givingThe fluctuation Euclidean action, the homogeneous boundary conditions, and the measure normalization contain neither $x_i$ nor $x_f$. Hence $\mathcal{N}$ is independent of both endpoint positions. It depends on $\beta ,m,\hslash$.

With the standard position-state normalization, the Euclidean free-particle bridge amplitude isFor example, a momentum resolution gives a one-dimensional Gaussian integral,whose evaluation gives the same exponent and prefactor. In particular, the result integrates to one over $x_f$ at fixed $x_i$, as required for its heat kernel interpretation.